Question

1. the percent of voters between the ages of 18 and 29 that participated in each united states presidential election between the years 1988 to 2016 are shown in the table.

year	1988	1992	1996	2000	2004	2008	2012	2016

the function p gives the percent of voters between 18 and 29 years old that participated in the election in year t.
a. determine the average rate of change for p between 1992 and 2000.
b. pick two different values of t so that the function has a negative average rate of change between the two values. determine the average rate of change.
c. pick two values of t so that the function has a positive average rate of change between the two values. determine the average rate of change.

Explanation:

Part (a)

Step1: Recall the formula for average rate of change

The average rate of change of a function \( y = P(t) \) between \( t = a \) and \( t = b \) is given by \( \frac{P(b)-P(a)}{b - a} \).

Step2: Identify the values for 1992 and 2000

For \( t = 1992 \), \( P(1992)=42.7 \) and for \( t = 2000 \), \( P(2000) = 34.5 \).

Step3: Substitute into the formula

We have \( a=1992 \), \( b = 2000 \), \( P(a)=42.7 \), \( P(b)=34.5 \).
The average rate of change \(=\frac{34.5 - 42.7}{2000-1992}=\frac{- 8.2}{8}=- 1.025\)

Part (b)

Step1: Choose two years with decreasing percentage

We choose \( t_1=1992 \) ( \( P(1992) = 42.7 \)) and \( t_2 = 1996 \) ( \( P(1996)=33.1 \))

Step2: Apply the average rate of change formula

The average rate of change \(=\frac{P(1996)-P(1992)}{1996 - 1992}=\frac{33.1-42.7}{4}=\frac{-9.6}{4}=- 2.4\)

Part (c)

Step1: Choose two years with increasing percentage

We choose \( t_1 = 2004 \) ( \( P(2004)=45.0 \)) and \( t_2=2008 \) ( \( P(2008) = 48.4 \))

Step2: Apply the average rate of change formula

The average rate of change \(=\frac{P(2008)-P(2004)}{2008 - 2004}=\frac{48.4 - 45.0}{4}=\frac{3.4}{4} = 0.85\)

Answer:

-1.025